Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is that every typable term (proof) reduces to a (necessarily unique) normal term (proof) We actually show a bit more: Proposition 4 (Strong Normalization) If M is typable, then it is strongly normalizing, i.e. every reduction sequence starting from M is nite. Proof. By the reducibility method [18]. Let SN be the set of strongly normalizing terms, and de ne an interpretation of types as sets of terms as follows: for every base type A, jjAjj =SN ; jjF Gjj =fM 2 SN jwhenever M x M 1 then for every N 2 jjF jj; M 1 fx : Ng 2 jjGjjg; jj F jj =fM 2 SN jwhenever M M 1 ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol. 7, Cambridge University Press, 1989.

.... dynamic semantics into a proof conditional fragment given by some rule set such as D ffi (or perhaps D 8 ) This would be surprising, given that proof conditional semantics can (under propositions astypes) be seen as a declarative (functional) programming language (e.g. Girard, Lafont and Taylor [10]) whereas dynamic semantics is commonly associated with assignment based (imperative) programs. Indeed, the thrust of (D) has more than once been identified with a procedural turn in semantics. Declarative though it may be, however, TTG is arguably more dynamic than dynamic semantics. ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1989.

....two parts, one managing the context and the other introducing the connective. This leads to formula transformations and proof transformations that preserve the elimination of main cuts. However, like CPS transformations, this mechanism eliminates non essential cuts known as commutative conversions [Gir87, GLT88]. A benefit of this strategy is to force cut elimination to be ChurchRosser. In fact there are two motivations for those transformations. The first one is that this method inhibits commutative conversion. As a consequence, this strategy can be compared to codings of linear logic into interaction ....

....our second order exponentials that look like the combinatorial approach of linear logic exponentials. This article presents successively the transformations for the additives Phi and , for exponentials and finish by constants. 2 Elimination of Additives 2. 1 Second Order Additives System F [GLT88] uses two kinds of sum type. The first version defines a new connective , rules to introduce and eliminate it and conversions of redexes. The second version emulates it as the abbreviation U V Def = 8X: U X) V X) X . Linear logic has the same property. We can use the additives Phi ....

Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

....of the properties of the artifacts that can exist inside computers: data and processes. Although the borderline to computing science may be fuzzy, we do count as subjects of computer science those of computability and complexity theory [70] automata theory and formal languages [62] type theory [44], foundations of algebraic semantics [9] denotational semantics [103] 4 , operational semantics [98] axiomatic semantics [60] and process algebras [59] Computer scientists create models of computing and students of computer science learn to solve equations 2.3 Computing Science ....

....3.6 Type Theory Type theory is one of the most important contributions that computer science has made to science. Starting with the implicit or explicit simple and ( trivially ) composite typing of variables and values of ordinary programming languages from Fortran to C, higher order types [44], such as of functions (which may again take on functions as parameters) subtypes ordered in lattices and used for example in object oriented systems (multiple inheritance etc. 24] and finally intuitionistic type theory [92] type theory is an exciting universe . Getting the types right ....

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Jean-Yves Girard, Y. Lafont, and P. Taylor. Proofs and Types, volume 7. Cambridge Univ. Press, Cambridge, UK, cambridge tracts in theoretical computer science edition, 1989.

....optimization possible from so called safe nodes. We conjecture that, with proper optimization, the bookkeeping is polynomial in the number of fan interactions. It also remains to be seen whether the expansion method could be generalized to higher order typed calculi, particularly System F [GLT89], which might give new and constructive proofs of strong normalization for those calculi. In summary, graph reduction for optimal evaluation presents a new technology for language implementation that is a hybrid of call by name and call by value. Its theoretical importance is unquestioned, and its ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

....as soon as we move to a (labelled) proof net approach. 3. 1 Labelled Gentzen Calculus The relations between Natural Deduction and sequent calculus for resource logics are well understood, syntactically and on the level of the Curry Howard interpretation, see for example [Gabbay de Queiroz, Girard e.a. 89, Wansing 92a] The move from Natural Deduction to Gentzen sequent presentation requires that we reformulate all logical rules of inference in such a way that a connective is introduced in the conclusion, either in the antecedent (rules of use, left rules) or in the succedent (rules of proof, ....

Jean-Yves Girard, Paul Taylor and Yves Lafont. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7, Cambridge, 1989.

....The interpretation of types is largely given in 6. We add that [ os B] B] This addition illuminates the semantics: for a computation type B, we can think of [ B] as the set of outsides for B. So, for example, the definition [ A B] A] Theta [ B] can be read in Heyting style [8]: an outside for A B is a value in A together with an outside for B. As usual a value Gamma v V : A denotes [ V ] Gamma ] Gamma [ A] and because a computation Gamma c M : B now has its outside as an extra parameter, it denotes [ M ] Gamma ] Theta [ A] Gamma 1. A ....

Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

....as soon as we move to a (labelled) proof net approach. 3. 1 Labelled Gentzen Calculus The relations between Natural Deduction and sequent calculus for resource logics are well understood, syntactically and on the level of the Curry Howard interpretation, see for example [Gabbay de Queiroz, Girard e.a. 89, Wansing 92a] The move from Natural Deduction to Gentzen sequent presentation requires that we reformulate all logical rules of inference in such a way that a connective is introduced in the conclusion, either in the antecedent (rules of use, left rules) or in the succedent (rules of proof, ....

Jean-Yves Girard, Paul Taylor and Yves Lafont. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7, Cambridge, 1989.

....the copying of existant redexes, the fi reduction of redexes with index oe can only construct new redexes of index oe or index . Normalization then follows by a straightforward induction on types alone. This contrasts with the familiar proof based on the idea of reducibility (see, e.g. [HS87, GLT89, Tai67]) 3 Sigma 0 is relative to Sigma if all redexes in Sigma 0 are in the same family of some redex in Sigma, see ( L evy78] Def. 4.4.1) 4 Simulating generic elementary time bounded computation Now that we know that the normal form of a simplytyped term can be computed in a linear ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

....corollary to confluency for MELL[2] In this paper, we focus on the intuitionistic version of decoration which is analogue of linear one. Our method can be seen as yet another proof of SN and CR property of cut elimination for LKT and LKQ, since typed calculus is also proven to be SN and CR [11]. Our contribution is to the notion how (intuitionistic decoration of) LKT and LKQ relate to the typed term assignments, hence LK; and to the observation that this is identical to CPS programs through the consideration on CND interpreted by LK. To our knowledge, this is the first paper that ....

....generates the CBV term assignment to (intuitionistic decoration of) LK. Theorem 1. Typed term assignment shown in Table 4 defines the SN and CR cut elimination procedure for LK. Proof. All of the propositions in this section and the fact that typed calculus is proven to be SN and CR [11]. Cut free LK derivation does not always mean cut free LKQ derivation, since (L ) in LK derivation converts into LKQ derivation including one additional m cut. This m cut is called correction cut. Eliminating of this m cut (i.e. contracting m cut redex) exactly means constrictive morphism[3] ....

Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

....operational semantics for the language may be obtained by directing the fi axioms from left to right. For more details about this system and the relation between its equational and operational semantics, consult [How92] related systems are considered many places in the literature, for example [GLT89, LS86, Mit90]. In a standard way, we may interpret a type expression oe containing a free type variable t as a functor; that is, it provides a map from types to types by substitution for t, and it may be extended to a map on terms of function type (because the intended model of this language is a cartesian ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

....procedure of MELL is proven to be SN and CR [3] Above properties for linear decoration is also true for intuitionistic decoration. Thus intuitionistic decoration can be seen as a part of yet another proof of SN and CR of LKT LKQ, since typed calculus is also proven to be SN and CR [12]. 2.4 Constrictive Morphisms Both LKT LKQ can be seen as restricted LK, thus it is sound w.r.t. classical provability. Simply ignoring semicolon from LKT LKQ derivation will induce LK derivation. LKT LKQ are also complete. DJS shows a method (constrictive morphism) through which one can convert ....

Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

....or the propositions as types paradigm. Logic provides not only basic input output specifications (i.e. types or formulas) but also a setting for well typed programs (i.e. terms or formal proofs) as well as a mode of execution of well typed programs by means of term reduction or normalization [2]. The advent of linear logic [3] with its intrinsic ability to reflect computational resources has made it possible to refine the propositions as types paradigm to computational complexity specifications. A bounded version of linear logic (BLL) was introduced in [4] in which the reuse of ....

..... Then 1 Contrary to [1] we do not assume x = x because this is not needed for the main features of LLL related to polynomial time. In particular, polynomial time functions are naturally represented in an intuitionistic version of LLL [1] which, as a type system, is a refinement of system F [13, 2]. by (Cl4) it suffices to show (f n (ff n 1 ) 1 n J n ) Delta (f n (fi n 1 ) 1 n J n ) Cl n (f n (ff n 1 fi n 1 ) 1 n J n ) Take an arbitrary element d from the left hand side. d is of the form f n (a) Delta f n (b) for some a 2 ff n 1 , b 2 fi n 1 . First, notice that f n ....

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Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

....interactions can be greatly reduced, or whether they are essential. A complementary tight upper bound on the number of sharing interactions has not been constructed. It also remains to be seen whether the j expansion method could be generalized to higher order typed calculi, particularly System F [GLT89], which might give new and constructive proofs of strong normalization for those calculi. Acknowledgements. For many relevant and fruitful discussions, suggestions, corrections, admonishments, and encouragements, we thank Arvind, Stefano Guerrini, Paris Kanellakis, Jakov Kucan, Julia Lawall, ....

Jean-Yves Girard, Yves Lafont, and Paul Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1989.

....calculus. In this paper, we focus on the intuitionistic version of decoration on which DJS do not put emphasis in [4] Otherwise said, our method can be seen as yet another proof of SN and CR property of cut elimination procedure for LKQ, since typed calculus is also proven to be SN and CR [11]. Our CPS calculi are constructed directly on a Gentzen style (intuitionistic decoration of) LKT and LKQ. Our first contribution is to the notion how (intuitionistic decoration of) LKQ precisely relate to the typed term assignments; and to the observation that this is identical to CBV CPS ....

....above automatically generates the CBV term assignment to (intuitionistic decoration of) LK(Table 4) Theorem 3.8 (LK as a Gentzen style type theory) Typed term assignment shown in Table 4 defines the SN and CR cut elimination procedure for LK. Proof. Typed calculus is proven to be SN and CR [11]. 2 Cut free LK derivation does not always produce cut free LKQ derivation. It is because that (L ) in LK derivation is converted into LKQ derivation with one additional m cut. 4 CPS transforms One may think that our term assignment for CBV CPS calculus is too limited. It obeys the restricted ....

Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

....] B The direction of cut is not specified in LK term calculus. So if we choose to erase the lhs of coequation, we get A x = 1 A ; A y = A as a result. Alternatively, if we choose to erase the rhs, we get A x = A ; A y = 2 A . This is exactly the one known as inconfluency phenomena[12](p.151) of classical proofs. Witness of peirce s law is the switch box of this inconfluency. The coloring t(CBN) or q (CBV) to the LK formula is the switch of this switch box, since coloring specifies the order and orientation of cut through the computational property of linear logic(i.e. linear ....

Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

....Cut elimination procedure of MELL is proven to be SN and CR [2] Above properties for linear decoration is also true for intuitionistic decoration. Thus intuitionistic decoration can be seen as yet another proof of SN and CR of LKT LKQ, since typed calculus is also proven to be SN and CR [11]. 2.3 Constrictive Morphisms Both LKT LKQ can be seen as restricted LK, thus it is sound w.r.t. classical provability. As ignoring semicolon from LKT LKQ derivation will induce LK derivation. LKT LKQ are also complete. DJS shows a method (constrictive morphism) through which one can convert ....

....of type :A. That is, he fix OE to A. Felleisen also use I instead of x:kx in CBV. It is known that calculus with non local exit operator C is inconfluent between CBN and CBV case. Our CPS calculus shows that peirce s law, in fact be an instance of popular example of LK is not constructive ([11] p.151) of the following form: B (RW) B ) LW) cut) Choosing CBN or CBV reduction scheme exactly means choosing orientation of cut. 3.8 Relation with Herbelin s calculus In Herbelin s work, terms are assigned to LJT which can be regarded as intuitionistic version of LKT ....

Jean-Yves Girard, Yves Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press, 1988.

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